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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2307.12955 |
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| _version_ | 1866913181381689344 |
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| author | Males, Joshua |
| author_facet | Males, Joshua |
| contents | In this short note, we prove equidistribution results regarding three families of three-colour partitions recently introduced by Schlosser and Zhou. To do so, we prove an asymptotic formula for the infinite product $F_{a,c}(ζ; {\rm e}^{-z}) := \prod_{n \geq 0} \big(1- ζ{\rm e}^{-(a+cn)z}\big)$ ($a,c \in \mathbb{N}$ with $0<a\leq c$ and $ζ$ a root of unity) when $z$ lies in certain sectors in the right half-plane, which may be useful in studying similar problems. As a corollary, we obtain the asymptotic behaviour of the three-colour partition families at hand. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_12955 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A note on the equidistribution of $3$-colour partitions Males, Joshua Combinatorics Number Theory In this short note, we prove equidistribution results regarding three families of three-colour partitions recently introduced by Schlosser and Zhou. To do so, we prove an asymptotic formula for the infinite product $F_{a,c}(ζ; {\rm e}^{-z}) := \prod_{n \geq 0} \big(1- ζ{\rm e}^{-(a+cn)z}\big)$ ($a,c \in \mathbb{N}$ with $0<a\leq c$ and $ζ$ a root of unity) when $z$ lies in certain sectors in the right half-plane, which may be useful in studying similar problems. As a corollary, we obtain the asymptotic behaviour of the three-colour partition families at hand. |
| title | A note on the equidistribution of $3$-colour partitions |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2307.12955 |